Preheat oven to 400°F (200°C). In a large saucepan, melt butter. Add 2 tbsp (30 mL) sugar and stir until smooth. Add apple and cook on medium-low heat until pieces are slightly softened but still hold their shape, 5 to 7 min. Stir in 1/8 tsp (0.5 mL) of cinnamon. Transfer to a bowl and set aside.
In the same saucepan, add apple cider, cider vinegar and remaining sugar and cinnamon and heat on medium until sugar dissolves. Turn heat to low and simmer until cider mixture is syrupy and reduced to about 3/4 cup (175 mL), approx 25 min.
Meanwhile, on a lightly floured surface, roll out puff pastry into a 14 x 14-in. (35 x 35 cm) square. Use a knife or pizza wheel to cut pastry into 12 rectangular pieces by slicing 4 horizontal rows and 3 vertical rows. Place cut pastry pieces into a greased 12-cup muffin tin, lining bottoms and sides. Use a fork to prick the bottoms a few times. Spoon cooked apple into pastry cups. Bake 15 min. or until golden brown. When cool enough to handle, remove tartlets from tin, drizzle with cider syrup and sprinkle with cheese. Serve hot or cold.